Two New Classes of Exact Solutions for the KdV Equation via Bgicklund Transformations-/- A. H. KHATERS, 0. H. EL-KALAAWY Department of Mathematics, Faculty of Science, Cairo Liniversity. Beni-Suef, Egypt and M. A. HELAL Department of Mathematics, Faculty of Science, Cairo University. Giza. Egypt (Accepted 3 April 1997) Abstract-The Korteweg-de Vries equation which includes nonlinear and dispersive terms quadratic in the wave amplitude is considered. The exact solutions can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system: that is. a linear eigenvaluc problem in the form of a system of first order partial differential equations. The method of characteristics is used and Backlund transformations (BTs) arc employed to generate two new solutions from the old one. 0 1997 Elscvier Science Ltd 1. INTRODUCTION The problem of wave propagation may be considered as the one which stands at the frontier of nonlinear domains in both mathematical and physical sciences (e.g. fluid dynamics, plasma physics, solid-state physics, optics, etc.). Indeed, it was the first to be observed, analysed and termed as solitary wave or soliton [l, 21. During the last three decades or so an exciting and extremely active area of research has been devoted to constructing exact solutions for a wide class of nonlinear equations. This includes the most famous nonlinear equation, the Korteweg-de Vries (KdV) equation. Indeed, various branches of mathematics as well as physics have been developed, renewed or even begun to meet the needs of the nonlinear world. Certain important and novel subareas of research lead to the construction of exact solutions, such as the applications of the Jacobian elliptic function [3], the powerful inverse scattering transform (IST) [4], the Backlund transformations (BTs) [5], the Painleve analysis [&lo], the Lie group theoretic methods [ll], the direct algebraic method [12-161 and tangent hyperbolic method [17-201. In this paper, two classes of new exact analytical solution are generated by using the BT technique for the KdV equation qr + 6~7, + qvrr = 0. (1) The outline is as follows. First, we address the Ablowitz, Kaup, Newell, Segur (AKNS) tcommunicated by Professor Wadati. fPresent address: Physics Department, U.I.A., University of Antwerp. B-2610 Antwerp, Belgium. 1901